Critical spin chains and loop models with $PSU(n)$ symmetry

TL;DR

The paper constructs and analyzes two-dimensional statistical models with PSU(n) symmetry, realized both as loop gases and alternating spin chains. By exploiting representation theory of U(n), Walled Brauer algebras, and diagram algebras (JTl), the authors derive the PSU(n) CFT spectrum, including the PSU(n) representations Omega_{(r,s)} and their decomposition under global symmetries, via twisted torus partition functions. They establish a detailed link between lattice algebras and continuum CFTs, and show that the O(n) CFT emerges as a Z2 orbifold of the PSU(n) CFT, with evidence from partition functions, fusion, and monodromies. The work provides a coherent algebraic and field-theoretic framework for PSU(n) loop/spin models, clarifies their phase structure, and connects to broader sigma-model and orbifold constructions, offering a pathway to non-integer n analyses and potential applications to CP^{n-1} theories.

Abstract

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the $O(n)$ model (symmetry group $O(n)$) and the Potts model (symmetry group $S_Q$). Both models make sense for $n,Q\in \mathbb{C}$ and not just $n,Q\in \mathbb{N}$, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group $PSU(n)$. We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT that exists for any $n\in\mathbb{C}$ and has a global $PSU(n)$ symmetry. Its spectrum is similar to those of the $O(n)$ and Potts CFTs, but a bit simpler. We conjecture that the $O(n)$ CFT is a $\mathbb{Z}_2$ orbifold of the $PSU(n)$ CFT, where $\mathbb{Z}_2$ acts as complex conjugation.